Supervised Deep Neural Networks (DNNs) for Pricing/Calibration of Vanilla/Exotic Options Under Various Different Processes
Supervised Deep Neural Networks (DNNs) for Pricing/Calibration of Vanilla/Exotic Options Under Various Different Processes
This paper demonstrates the application of supervised deep neural networks (DNNs) for pricing and calibration of vanilla and exotic options under various stochastic processes, including geometric Brownian motion (GBM), variance gamma (VG), and their stochastic volatility counterparts (GBMSA and VGSA). The authors generate training labels using traditional methods (closed-form solutions, FFT, Monte Carlo) and train feedforward networks with different architectures. They find that a network with 4 hidden layers, 120 neurons per layer, and leaky ReLU activation functions, trained with the Adam optimizer, provides the best balance of accuracy and speed. The DNNs achieve pricing speeds orders of magnitude faster than traditional methods, with mean squared errors typically below 0.0001 and R² values above 99% for in-sample tests. The method also generalizes well to out-of-sample scenarios, including deep out-of-the-money options and longer maturities. The authors conclude that DNNs are a practical tool for real-time pricing and calibration, with accuracy within typical bid-ask spreads.
Highlights
- 1Deep neural networks (DNNs) achieve exponential speedup in option pricing compared to traditional methods like Monte Carlo and FFT.
- 2The study covers vanilla and exotic options (European, Barrier, American) under four processes: GBM, VG, GBMSA, and VGSA.
- 3Optimal DNN architecture found: 4 hidden layers with 120 neurons each and leaky ReLU activation functions.
- 4Training with Halton sequences (quasi-random) yields slightly better performance than uniform random sampling.
- 5The method enables super-fast calibration and parameter estimation, with accuracy within bid-ask spreads.
Methods
- MSupervised feedforward neural networks with various architectures (layers, neurons, activation functions).
- MLabel generation using closed-form solutions (Black-Scholes), FFT, Monte Carlo simulation, and Ju-Zhong approximation.
- MOptimization with Adam optimizer and MSE loss function.
- MHyperparameter tuning: number of layers (2-6), neurons per layer (up to 200), activation functions (ReLU, leaky ReLU, ELU, sigmoid).
Results
- RFor European options under GBM, MSE as low as 0.000018 and R² up to 99.79% with 240,000 training samples.
- RBarrier option pricing under GBM achieved MSE of 0.000016 and R² of 99.57% with 240,000 samples.
- RAmerican option pricing under GBM showed MSE of 0.000022 and R² of 99.72% with 240,000 samples.
- RPerformance degrades with more complex models (VGSA) but remains high (R² > 99% for in-sample).
- RExtrapolation to deep out-of-the-money and longer maturities shows good generalization, with R² often above 98%.
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